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<p>The characteristic equation of (<a href="" class="xref" data-knowl="./knowl/eq5_23.html" title="Equation 5.4.2">(5.4.2)</a>) is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_23.html">
\begin{equation*}
r^2+(\alpha-1)r+\beta=0.
\end{equation*}
</div>
<p class="continuation">Real and distinct roots <span class="process-math">\(r_1, r_2\text{:}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_23.html">
\begin{equation}
y=C_1 e^{r_1z}+C_2 e^{r_2 z}=C_1 e^{r_1 \ln x}+C_2 e^{r_2 \ln x}=C_1 x^{r_1}+C_2 x^{r_2}.\tag{5.4.3}
\end{equation}
</div>
<p class="continuation">Equal roots <span class="process-math">\(r_1\)</span> and <span class="process-math">\(r_2\text{:}\)</span></p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_23.html">
\begin{equation}
y=C_1 e^{r_1 z}+C_2 z e^{r_1 z}=C_1 x^{r_1}+C_2 \ln x \cdot x^{r_1}.\tag{5.4.4}
\end{equation}
</div>
<p class="continuation">Complex roots: (<span class="process-math">\(r_{1, 2}=\lambda \pm i \mu\)</span>)</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq5_23.html">
\begin{equation*}
y=C_1 e^{\lambda z} \cos (\mu z)+C_2 e^{\lambda z} \sin (\mu z)=C_1 e^{\lambda \ln x} \cos (\mu \ln x)+C_2 e^{\lambda \ln x} \sin (\mu \ln x).
\end{equation*}
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